Schrödinger's Equation and Quantum Mechanics

Nielsen, M. P., Laugesen, Jacob, Sanjayan, J. G.

Jing‐Bo Chen

Multisymplectic geometry for the Schrödinger equation in quantum mechanics is presented. This formalism of multisymplectic geometry provides a concise and complete introduction to the Schrödinger equation. The Schrödinger equation, its associated energy and momentum evolution equations, and the multisymplectic form are derived directly from the variational principle. Some applications are also explored.

V. Milos, V. Kundrát, Jiri Prochazk

Yisong Yang

Abstract The essence of quantum mechanics is the Schrödinger equation. Thus, the path to quantum mechanics lies in an understanding of the birth of the Schrödinger equation. This chapter derives the Schrodinger equation based on the photoelectric effect, which has found a wide range of applications in today’s electronics. It then discusses the Schrodinger equation’s statistical interpretations and applications. It goes on to present the quantum many-body problem, which is of contemporary research interest, as well as its many applications. Next it investigates the Hartree-Fock method and the Thomas-Fermi approach, as well as provides a discussion of the Rydberg formula. Finally, the chapter looks at and unpacks the density functional theory.

Nima Moshayedi

C.-L. Ho, Ryu Sasaki B

Based on the well-known relation between Fokker-Planck equations and Schrödinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schrödinger equations of “discrete” QM. The latter is a natural discretization of QM and its Schrödinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable “discrete”QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.